Approaching the thermodynamic limit in equilibrated scale-free networks.

We discuss how various models of scale-free complex networks approach their limiting properties when the size N of the network grows. We focus mainly on equilibrated networks and their finite-size degree distributions. Our results show that the position of the cutoff in the degree distribution, k_{cutoff} , scales with N in a different way than predicted for N-->infinity ; that is, subleading corrections to the scaling k_{cutoff} approximately N;{alpha} are strong even for networks of order N approximately 10;{9} nodes. We observe also a logarithmic correction to the scaling for degenerated graphs with the degree distribution pi(k) approximately k;{-3} . On the other hand, the distribution of the maximal degree k_{max} may have a different scaling than the cutoff and, moreover, it approaches the thermodynamic limit much faster. We argue that k_{max} approximately N;{alpha;{'}} with an exponent alpha;{'}=min[alpha,1(gamma-1)] , where gamma is the exponent in the power law pi(k) approximately k;{-gamma} . We also present some results on the cutoff function and the distribution of the maximal degree in equilibrated networks.

Condensation in zero-range processes on inhomogeneous networks.

We investigate the role of inhomogeneities in zero-range processes in condensation dynamics. We consider the dynamics of balls hopping between nodes of a network with one node of degree k_{1} much higher than a typical degree k , and find that the condensation is triggered by the inhomogeneity and that it depends on the ratio k_{1}k . Although, on the average, the condensate takes an extensive number of balls, its occupation can oscillate in a wide range. We show that in systems with strong inhomogeneity, the typical melting time of the condensate grows exponentially with the number of balls.

Balls-in-boxes condensation on networks.

We discuss two different regimes of condensate formation in zero-range processes on networks: on a q-regular network, where the condensate is formed as a result of a spontaneous symmetry breaking, and on an irregular network, where the symmetry of the partition function is explicitly broken. In the latter case we consider a minimal irregularity of the q-regular network introduced by a single Q node with degree Q>q. The statics and dynamics of the condensation depend on the parameter alpha=ln Q/q, which controls the exponential falloff of the distribution of particles on regular nodes and the typical time scale for melting of the condensate on the Q node, which increases exponentially with the system size N. This behavior is different than that on a q-regular network, where alpha=0 and where the condensation results from the spontaneous symmetry breaking of the partition function, which is invariant under a permutation of particle occupation numbers on the q nodes of the network. In this case the typical time scale for condensate melting is known to increase typically as a power of the system size.